Rate of Change and Slope

In this lesson you will: Compute slope as the change in output divided by the change in input; Find a rate of change from two points, from a table, or from a graph; Attach units to a slope and say what the number means in the situation.

Steepness is a number, not a feeling

Take any two points on a line. Divide the change in the output by the change in the input. That quotient is the slope, and for a line it comes out the same no matter which two points you picked.

People say \"rise over run\" because the top of the fraction is vertical movement and the bottom is horizontal movement. The word you want in a context is rate of change: dollars per hour, gallons per minute, degrees per day.

Measure anywhere, get the same answer

This is the property that makes a line a line. Move four across and three up from one place, and you are on the graph again; do it from somewhere else on the line and you are still on the graph.

That constancy is also the test for linearity in a table: equal steps in the input must produce equal steps in the output.

Four things a slope can be

A line that climbs from left to right has positive slope. One that falls has negative slope. A horizontal line goes nowhere vertically, so its rise is 0 and its slope is 0.

A vertical line is the odd one. Its run is 0, and the slope formula would ask you to divide by zero, so a vertical line has no slope — which is not the same as having a slope of zero.

Common mistake — Subtracting the coordinates in different orders

For (2,5) and (6,17) a student writes m=(17-5)/(2-6)=(12)/(-4)=-3.

The numbers used are all correct numbers. The problem is bookkeeping: the top subtracted the second point's output first, and the bottom subtracted the first point's input first. Mixing the order flips the sign of the answer and nothing on the page warns you.

Done consistently, (17-5)/(6-2)=3 or (5-17)/(2-6)=(-12)/(-4)=3. Both orders are fine; mixing them is not.

How to catch it: Sketch the two points, even roughly. If the second point is up and to the right of the first, the slope has to be positive, so a negative answer means the subtraction order got crossed.